RECORDING Jane's band wants to spend no more than 575...

RECORDING Jane's band wants to spend no more than $\$ 575$ recording their first $C D$ . The studio charges at least $\$ 35$ an hour to record. Graph a system of inequalities to represent this situation.

Key Concepts

Budget Constraints

Budget constraints are used to express limitations on spending in mathematical problems, typically through inequalities. They represent the maximum or minimum amounts available or required, allowing the setup of conditions where the total expenditure must not exceed (or must be at least) a certain amount.

System of Inequalities

A system of inequalities consists of two or more inequalities that are considered together. The solution to the system is the set of all points that simultaneously satisfy every inequality, representing multiple constraints that must all be met.

Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols (such as ?, ?, <, >) instead of an equals sign. They represent constraints in which one side of the inequality is bound by a linear function, creating a region in the coordinate plane rather than a single line.

Graphing Inequalities

Graphing inequalities involves drawing the boundary line corresponding to the equality part of the inequality and then shading the region that satisfies the inequality. A solid line is used when the inequality includes equality (? or ?), and a dashed line is used when it does not (< or >).

Feasible Region

The feasible region is the overlapping area on a graph that satisfies all the inequalities in a system simultaneously. It represents all the possible solutions under the given constraints, such as budget limits or minimum requirements.

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